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• W1996965489 abstract "Abstract The algebraic matrix Riccati equation AX + XA T − XFX + C =0, where matrices A, B, C, F ∈ ℝ n × n are given and a solution X ∈ ℝ n × n is sought, plays a fundamental role in optimal control problems. Large‐scale systems typically appear if the constraint is described by a partial differential equation (PDE). We provide a nonlinear multigrid algorithm that computes the solution X in a data‐sparse, low‐rank format and has a complexity of 𝒪( n ), subject to the condition that F and C are of low rank and A is the finite element or finite difference discretization of an elliptic PDE. We indicate how to generalize the method to ℋ︁‐matrices C, F and X that are only blockwise of low rank and thus allow a broader applicability with a complexity of 𝒪( n log( n ) p ), p being a small constant. The method can also be applied to unstructured and dense matrices C and X in order to solve the Riccati equation in 𝒪( n 2 ). Copyright © 2008 John Wiley & Sons, Ltd." @default.
• W1996965489 created "2016-06-24" @default.
• W1996965489 creator A5044715882 @default.
• W1996965489 date "2008-06-19" @default.
• W1996965489 modified "2023-09-25" @default.
• W1996965489 title "Nonlinear multigrid for the solution of large-scale Riccati equations in low-rank and ℋ︁-matrix format" @default.
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• W1996965489 doi "https://doi.org/10.1002/nla.606" @default.
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